Neural Computation 0368-4149-01

Prof. Nathan IntratorTuesday 16:00-19:00 Schreiber 007

Office hours: Wed 4-5nin@tau.ac.il

Outline

• Goals for neural learning - Unsupervised• Goals for statistical/computational learning

– PCA– ICA– Exploratory Projection Pursuit– Search for non-Gaussian distributions

• Practical implementations

Statistical Approach to Unsupervised Learning

• Understanding the nature of data variability• Modeling the data (sometimes very flexible model)• Understanding the nature of the noise• Applying prior knowledge• Extracting features based on:

– Prior knowledge– Class prediction– Unsupervised learning

4

Principal Component Analysis.

Włodzisław Duch

SCE, NTU, Singapore

http://www.ntu.edu.sg/home/aswduch

Linear transformations – example

2D vectors X in a unit circle with mean (1,1); Y = A*X, A = 2x2 matrix

The shape is elongated, rotated and the mean is shifted.

1 1

2 2

2 1

1 1

Y X

Y X

Invariant distances

Euclidean distance is not invariant to general linear transformations

This is invariant only for orthonormal matrices ATA = I that make rigid rotations, without stretching or shrinking distances.

Idea: standardize the data in some way to create invariant distances.

Y A X

T21 2 1 2 1 2

T1 2 1 2T

Y Y Y Y Y Y

X X A A X X

Data standardization

For each vector component X(j)T=(X1(j), ... Xd

(j)), j=1 .. n

calculate mean and std: n – number of vectors, d – their dimension

( ) ( )

1 1

1 1;

i

n nj j

ij j

X Xn n

X XVector of mean

feature values.

Averages over rows. (1) (2) ( )

(1) (2) ( )1 1 1 1

(1) (2) ( )2 2 2 2

(1) (2) ( )

n

n

n

nd d d d

X X X X

X X X X

X X X X

X X X

Standard deviation

Calculate standard deviation:

Transform X => Z, standardized data vectors

( )

1

22 ( )

1

1

1

1

i

i i

nj

ij

nj

ij

X Xn

X Xn

Vector of mean feature values.

Variance = square of standard deviation (std), sum of all deviations from the mean value.

( ) ( )j ji i i iZ X X

Std data

Std data: zero mean and unit variance.

Standardize data after making data transformation.

Effect: data is invariant to scaling only (diagonal transformation).

Distances are invariant, data distribution is the same.

How to make data invariant to any linear transformations?

,

( ) ( )

1 1

2 22 ( ) ( ) 2

1 1

1 10

1 11

1 1

i i

Z i i i

n nj j

i i ij j

n nj j

i i ij j

Z Z X Xn n

Z Z X Xn n

Data standardization example

For our example Y=AX, assuming X means=1 and variances = 1

Transformation

Vector of mean

feature values.

Variance

check it!

1 3 2 1 1

1 2 1 1 1a

X Y

1 1

2 2

2 1

1 1

Y X

Y X

T1 5Diag

1 2

X Yσ σ AA

T21 2 1 2 1 2T Y Y X X A A X X How to make this

invariant?

Covariance matrixVariance (spread around mean value) + correlation between features.

where X is d x n dimensional matrix of vectors shifted to their means.

Covariance matrix is symmetric Cij = Cji and positive definite.

Diagonal elements are variances (square of std), si2 = Cii

( ) ( )

1

T( ) ( ) T

1

1; , 1

1

1 1

1 1

i

nk k

ij i j jk

nk k

k

C X X X X i j dn

n n

XC X X X X XX

[ 1, 1]ij ij i jr C Spherical distribution of data has Cij=I (unit matrix).

Elongated ellipsoids: large off-diagonal elements, strong correlations between features.

CX is d x d

Mahalanobis distance

Linear combinations of features leads to rotations and scaling of data.

Mahalanobis distance:

is invariant to linear transformations:

T; ; Y X Y AX Y AX C AC A

T21 2 1 2 1 21

T 11 2 1 2T T 1 1

21 2

Y

X

YC

X

C

Y Y Y Y C Y Y

X X A A C A A X X

X X

2 T 1

XXC

X X C X

Principal componentsHow to avoid correlated features?

Correlations covariance matrix is non-diagonal !

Solution: diagonalize it, then use transformation that makes it

diagonal to de-correlate features.

C – symmetric, positive definite matrix XTCX > 0 for ||X||>0;

its eigenvectors are orthonormal:

its eigenvalues are all non-negative

Z – matrix of orthonormal eigenvectors (because Z is real+symmetric),

transforms X into Y, with diagonal CY, i.e. decorrelated.

T ( ) ( )

T T

; ;i ii

X X

Y X

Y Z X C Z Z C Z ZΛ

C Z C Z Z ZΛ Λ

In matrix form, X, Y are dxn, Z, CX, CY are dxd

( )T ( )i jij Z Z

Matrix form

Eigen problem for C matrix in matrix form: X aC Z ZΛ

11 12 1 11 12 1

21 22 2 21 22 2

1 2 1 2

11 12 1 1

21 22 2 2

1 2

0 0

0 0

0 0

d d

d d

d d dd d d dd

d

d

d d dd d

C C C Z Z Z

C C C Z Z Z

C C C Z Z Z

Z Z Z

Z Z Z

Z Z Z

Principal componentsPCA: old idea, C. Pearson (1901), H. Hotelling 1933

Result: PC are linear combinations of all features, providing new uncorrelated features, with diagonal covariance matrix = eigenvalues.

T

T

;

Y X

Y Z X

C Z C Z Λ

TXZΛZ C

Small li small variance data change little in direction Yi

PCA minimizes C matrix reconstruction errors:

Zi vectors for large li are sufficient to get:

because vectors for small eigenvalues will have very

small contribution to the covariance matrix.

Y – principal components, or vectors X transformed using eigenvectors of CX

Covariance matrix of transformed vectors is diagonal => ellipsoidal distribution of data.

Two components for visualization

New coordinate system: axis ordered according to variance = size of the eigenvalue.

First k dimensions account for

1

1

k

ii

dk

ii

V

fraction of all variance (please note that li are variances); frequently 80-90% is sufficient for rough description.

Diagonalization methods: see Numerical Recipes, www.nr.com

PCA properties

PC Analysis (PCA) may be achieved by:

• transformation making covariance matrix diagonal

• projecting the data on a line for which the sums of squares of distances from original points to projections is minimal.

• orthogonal transformation to new variables that have stationary variances

True covariance matrices are usually not known, estimated from data.

This works well on single-cluster data; more complex structure may require local PCA, separately for each cluster.

PC is useful for: finding new, more informative, uncorrelated features;

reducing dimensionality: reject low variance features,

reconstructing covariance matrices from low-dim data.

PCA Wisconsin exampleWisconsin Breast Cancer data:

• Collected at the University of Wisconsin Hospitals, USA.

• 699 cases, 458 (65.5%) benign (red), 241 malignant (green).

• 9 features: quantized 1, 2 .. 10, cell properties, ex:

Clump Thickness, Uniformity of Cell Size, Shape, Marginal Adhesion, Single Epithelial Cell Size, Bare Nuclei,

Bland Chromatin, Normal Nucleoli, Mitoses.

2D scatterograms do not show any structure no matter which subspaces are taken!

Example cont.PC gives useful information already in 2D.

Taking first PCA component of the standardized data:

If (Y1>0.41) then benign else malignant

18 errors/699 cases = 97.4%

Transformed vectors are not

standardized, std’s are below.

Eigenvalues converge slowly, but classes are

separated well.

PCA disadvantages

Useful for dimensionality reduction but: • Largest variance determines which components are used, but

does not guarantee interesting viewpoint for clustering data.• The meaning of features is lost when linear combinations are

formed.

Analysis of coefficients in Z1 and other important eigenvectors may show which original features are given much weight.

PCA may be also done in an efficient way by performing singular value decomposition of the standardized data matrix.

PCA is also called Karhuen-Loève transformation.

Many variants of PCA are described in A. Webb, Statistical pattern recognition, J. Wiley 2002.

2 skewed distributions

PCA transformation for 2D data:

First component will be chosen along the largest variance line, both clusters will strongly overlap, no interesting structure will be visible.

In fact projection to orthogonal axis to the first PCA component has much more discriminating power.

Discriminant coordinates should be used to reveal class structure.

High Dimensional Data

Dimension Reduction

Feature ExtractionVisualisationClassification

Analysis

Projection Pursuit

what: An automated procedure that seeks interesting low dimensional projections of a high dimensional cloud by numerically maximizing an objective function or projection index.

Huber, 1985

Projection Pursuitwhy:

Curse of dimensionality

• Less Robustness

• worse mean squared error

• greater computational cost

• slower convergence to limiting distributions

• …

• Required number of labelled samples increases with dimensionality.

What is an interesting projection

In general: the projection that reveals more

information about the structure.

In pattern recognition:

a projection that maximises class separability in a low dimensional

subspace.

Projection Pursuit

Dimensional ReductionFind lower-dimensional projections of a high-dimensional point cloud to facilitate

classification.

Exploratory Projection PursuitReduce the dimension of the problem to facilitate visualization.

Projection Pursuit

How many dimensions to use

• for visualization

• for classification/analysis

Which Projection Index to use

• measure of variation (Principal Components)

• departure from normality (negative entropy)

• class separability(distance, Bhattacharyya, Mahalanobis, ...)

• …

Projection Pursuit

Which optimization method to choose

We are trying to find the global optimum among local ones

• hill climbing methods (simulated annealing)

• regular optimization routines with random starting points.

Timetable for Dimensionality reduction

• Begin 16 April 1998

• Report on the state-of-the-art. 1 June 1998

• Begin software implementation 15 June 1998

• Prototype software presentation 1 November 1998